IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 3, MARCH 2011 1217

Exploiting Long-Term Channel Correlationin Limited Feedback SDMA Through

Channel Phase CodebookYongming Huang, Member, IEEE, Luxi Yang, Member, IEEE, Mats Bengtsson, Senior Member, IEEE, and

Björn Ottersten, Fellow, IEEE

Abstract—Improving channel information quality at the basestation (BS) is crucial for the optimization of frequency divisionduplexed (FDD) multi-antenna multiuser downlink systems withlimited feedback. To this end, this paper proposes to estimate aparticular representation of channel state information (CSI) atthe BS through channel norm feedback and a newly developedchannel phase codebook, where the long-term channel correlationis efficiently exploited to improve performance. In particular, thechannel representation is decomposed into a gain-related part anda phase-related part, with each of them estimated separately. Morespecifically, the gain-related part is estimated from the channelnorm and channel correlation matrix, while the phase-relatedpart is determined using a channel phase codebook, constructedwith the generalized Lloyd algorithm. Using the estimated channelrepresentation, joint optimization of multiuser precoding and op-portunistic scheduling is performed to obtain an SDMA transmitscheme. Computer simulation results confirm the advantage ofthe proposed scheme over state of the art limited feedback SDMAschemes under correlated channel environment.

Index Terms—Limited feedback, multi-antenna multiuserdownlink, phase codebook, SDMA, user scheduling.

Manuscript received August 14, 2009; revised March 24, 2010, September30, 2010; accepted October 31, 2010. Date of publication November 22, 2010;date of current version February 09, 2011. The associate editor coordinatingthe review of this manuscript and approving it for publication was Dr. MartinSchubert. This work was supported in part by the NBRPC/973 under Grant2007CB310603, the National Science and Technology Major Project of Chinaunder Grants 2011ZX03003-001 and 2011ZX03003-003, the NSFC underGrants 60902012 and 61071113, the European Research Council under theEuropean Community’s Seventh Framework Programme (FP7/2007-2013) andERC grant agreement no. 228044, the Ph.D. Programs Foundation of Ministryof Education of China under Grants 20090092120013 and 20100092110010,and the Huawei Technologies Corporation. Part of this work was conductedwhen Y. Huang was visiting ACCESS Linnaeus Center, KTH Signal ProcessingLab, Royal Institute of Technology, SE-100 44 Stockholm, Sweden.

Y. Huang and L. Yang are with the School of Information Science and En-gineering, Southeast University, Nanjing 210096, China, and also with the KeyLaboratory of Underwater Acoustic Signal Processing of Ministry of Education,Southeast University, Nanjing 210096, China (e-mail: huangym@seu.edu.cn;lxyang@seu.edu.cn).

M. Bengtsson is with the ACCESS Linnaeus Center, KTH Signal ProcessingLab, Royal Institute of Technology, SE-100 44 Stockholm, Sweden (e-mail:mats.bengtsson).

B. Ottersten is with the ACCESS Linnaeus Center, KTH Signal ProcessingLab, Royal Institute of Technology, SE-100 44 Stockholm, Sweden, andalso with securityandtrust.lu, University of Luxembourg (e-mail: bjorn.otter-sten@ee.kth.se).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSP.2010.2094190

I. INTRODUCTION

I N multi-antenna multiuser downlink systems, space divi-sion multiple access (SDMA) may achieve a much larger

system throughput compared to conventional time divisionmultiple access (TDMA). The optimal capacity performanceof SDMA can be achieved by a multiuser precoding schemecalled dirty paper coding (DPC) [1]. In particular, when thenumber of users exceeds the number of transmit antennas

at base station (BS), a linear increase of the capacity withcan be achieved by DPC. However, in practical systems,

DPC is difficult to implement due to intensive computationalcomplexity especially when the number of users is large, andsensitivity to channel knowledge. To reduce the complexity,several suboptimal schemes have been developed recently,including the zero-forcing (ZF) precoding [2], block diago-nalized precoding [3], MMSE precoding [4], the generalizedeigenvalue based solution [5] and the iterative method basedsolution [6] etc. These low-complexity schemes can achievea large portion of the DPC capacity, but in the case of a largenumber of users they should be jointly designed with userscheduling algorithms [7], [8].

The main problem for the schemes above is that they all re-quire full channel state information (CSI) of all the users atthe base station. This is difficult to realize in practical systems.In frequency division duplexed (FDD) systems, CSI can typi-cally be estimated only at receivers and must be fed back to thebase station if needed. Unfortunately, the bandwidth of practicalfeedback channels is usually limited and must be shared by allthe users. Thus, in most cases, only very limited channel infor-mation can be obtained at the base station. To address this issue,a number of SDMA schemes based on limited feedback havebeen proposed recently. In particular, a limited feedback SDMAscheme based on opportunistic scheduling [9] has been shownto asymptotically achieve the optimal capacity scaling when theuser number tends to infinity. However, in networks where thenumber of users is limited, the performance of this scheme de-grades severely due to excessive mutual interference betweenmultiple simultaneously active users. Later, a kind of improvedSDMA scheme [10], [11] that uses a properly designed SDMAprecoding codebook, instead of using a random unitary matrixas the precoder, is proposed to handle this problem. Alterna-tively, the method of CSI quantization can be used to reduce thefeedback burden [12]–[14], which is effective especially wheneach user terminal is equipped with a single antenna.

1053-587X/$26.00 © 2010 IEEE

1218 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 3, MARCH 2011

In this paper, we focus on a multiuser downlink system de-sign in a wide area scenario or metropolitan scenario, where thechannel fading of different transmit antennas at base station isusually correlated due to the elevated antennas. Compared to theinstantaneous CSI, the long-term channel correlation changesmuch slower. So it can be estimated at the base station or beconveyed from the user terminal to the base station. It is easyto understand that combining this statistical information withlimited feedback in the design of SDMA schemes promises toimprove performance. This idea was first realized in the designof point-to-point MIMO precoding systems [15], and was thenextended to multiuser downlink systems. Typically, a maximumlikelihood (ML) estimate of the channel vector from channelspatial correlation and feedback channel norm, is proposed in[16], [17] for user scheduling. Later, it is shown in [18] that addi-tional feedback of a channel direction alignment parameter canrealize a low complexity user scheduling and beamforming. Al-ternatively, a minimum mean square error (MMSE) estimate ofa special form of CSI is proposed in [19] and [20], which is, ingeneral, more accurate than the ML estimate and can be used forthe joint design of user scheduling and multiuser precoding [21].However, this MMSE estimate heavily depends on the spatialcorrelation level, and its performance degrades quickly with thecorrelation level decreasing.

In contrast to the schemes above, here we devise a newmethod to exploit long-term channel correlation in the limitedfeedback SDMA scheme. To be specific, we propose to estimatethe channel representation (where denotes the channelvector) at the BS from newly designed feedback parameters andchannel correlation matrices, where the feedback parametersof each user consist of the channel norm and the index ofthe preferred codeword in a properly designed channel phasecodebook. By combining these feedback parameters with thechannel correlation matrix, an estimation method is furtherproposed to reconstruct the channel representation at theBS. The estimated channel representation is then used for userscheduling and multiuser precoding. It should be noted thatthough the optimal phase codebook construction is user depen-dent since it is based on channel correlation condition, a uniquephase codebook can be designed offline for different channelcorrelation conditions, then we adapt it to the user-specificchannel correlation condition using a unitary transformation,thus the resulting codebook is able to efficiently quantize thechannel phase matrix.

The rest of this paper is organized as follows. In Section II,we introduce the system model. In Section III we propose afeedback method and a related channel reconstruction algo-rithm to acquire channel knowledge at the BS, and generalizeit from single-antenna receivers to multi-antenna receivers. InSection IV, we use the generalized Lloyd algorithm to constructa novel channel phase codebook to complete the channel recon-struction algorithm. In Section V, we theoretically evaluate theperformance of the proposed channel reconstruction method.In Section VI, we use the reconstructed channel knowledgeat the BS to complete the user scheduling and transmit beam-forming design. Simulation results are given in Section VII,and conclusions are drawn in Section VIII.

II. SYSTEM MODEL

We consider a multi-antenna multiuser downlink systemwith one base station and users, in which the base stationis equipped with transmit antennas and each user terminalis equipped with a single receive antenna. In each block, thebase station selects out of users and communicates si-multaneously with them using the same frequency band. Wedenote the information symbols intended for the selectedusers as , which will be left-multiplied witha multiuser precoder, denoted as , beforesimultaneous transmission. The equivalent baseband receivesignal at the th user terminal can be written as

(1)

where denotes the transmitted signal vector from thebase station, denotes the additive white Gaussian noise withvariance , and denotes the downlink channelof the selected user .

In the above system model, the number of selected users ineach block is typically more than one, i.e., in an SDMA fashion.The information symbols are assumed to have unit averagepower. For simplicity, we also assume a uniform power allo-cation among users, though our scheme is not only limited tothis case. This means that the beam included in the mul-tiuser precoder is constrained with , where

indicates the total power constraint at the base station. Weconsider a wide-area or metropolitan scenario and assume thechannel vector obeys the distribution of , where

denotes the channel correlation matrix and is usually dom-inated by a small number of eigendirections.

To perform user scheduling and multiuser precoder design,the base station needs some channel knowledge. In practicalFDD systems, only partial CSI can be fed back from the userterminal to the base station via a finite-rate feedback channel.Besides, the long-term channel correlation information can alsobe obtained at the base station by a direct estimation from the re-verse link or alternatively through a feedback channel, due to thefact that it varies slowly. It is useful to combine the short-termpartial CSI and long-term channel correlation in the SDMAtransmit optimization. In this paper we aim to develop a methodto efficiently exploit the channel correlation in limited feedbackSDMA scheme and expect better performance than related ex-isting methods.

III. FEEDBACK PARAMETER AND CHANNEL

RECONSTRUCTION AT BS

In this section, we will devise a method to reconstruct thedownlink channel at the BS based on limited feedback and long-term channel correlation information, by defining the feedbackparameters and developing the channel reconstruction algorithmat the BS.

A. Feedback Parameters

Typically, feedback parameters from each user in limitedfeedback SDMA schemes consist of a channel quality indi-

HUANG et al.: EXPLOITING LONG-TERM CHANNEL CORRELATION IN LIMITED FEEDBACK SDMA 1219

cator (CQI) and a channel direction/phase indicator (CDI),where the CDI is usually generated by constructing a channelvector quantization codebook. To exploit long-term channelcorrelation if available, the quantization codebook should beconstructed based on correlated channel model [11], [14], [22]or be adaptively updated according to correlation matrices [23].Alternatively, based on a random unitary codebook, conveyingof an channel direction alignment parameter is used as CDIfeedback [17]. Then, by combining this short-term channelfeedback with long-term channel correlation, the user channelvector is estimated at the BS with maximum likelihood (ML)criterion. In contrast to these methods, here we propose touse the channel norm (the subscript is omittedfor notation simplicity) as CQI feedback and devise a channelphase codebook to generate a CDI feedback, based on whicha particular representation of user channel can be estimated orreconstructed at the BS, where the long-term channel correla-tion is exploited. In particular, our channel phase codebook isdefined as a set of unitary matrices which distinguishes it frommost existing schemes.

B. Channel Reconstruction Algorithm

In order to perform user scheduling and multiuser precoderdesign, the base station needs to reconstruct a certain formof CSI using the feedback information as well as the channelcorrelation matrix. Since we focus on environments where thechannel vector typically is zero-mean, it is not effective todirectly estimate using an MMSE or ML criterion. As shownin [16] and [17], the ML estimate of in general can only beused for user scheduling. To guarantee the performance, themultiuser precoder needs to be optimized based on full CSI ofthe selected users. Here, we propose to reconstruct the channelrepresentation instead of , which, as shown in thefollowing sections, is sufficient for the user scheduling andprecoding design. Furthermore, the estimate of in our paperwill be expressed in a form similar to the eigendecomposition,given by

(2)

where is a complex-valued matrix withorthonormal columns which represent the phase information ofchannel vector, and is a real-valueddiagonal matrix and represents the channel gain.

Note that strictly is a rank-one matrix and the straightfor-ward solution for its estimation suggests that is constrainedto include a single non-zero element. However, it may sacrifice

the robustness against some extreme channel realizations and re-sult in a limited performance, especially when the number of thefeedback bits is limited. Therefore, we would like to reconstruct

via a superposition of multiple eigenmodes , inwhich the first eigenmode is dominant over the others by

.Based on the form (2), we will derive the channel reconstruc-

tion algorithm starting with the eigendecomposition of

(3)

where denotes the eigenvectors of , and denotes thediagonal matrix with elements being the eigenvalues of

arranged in a non-increasing order. Using the channel normavailable at the base station, the eigenvalue related matrix

in our proposed method is calculated as

(4)

where denotes the expectation operator. The right sideof this equation can be viewed as an MMSE estimate of

, and it was proved in [19] and [20] that thisexpression is real-valued, diagonal and has its elements repre-senting the gains of the multiple eigenmodes of the channel.The reason that we choose this expression of is also justifiedin the performance analysis in Section V, where the proposedchannel reconstruction method of (2) and (4) will be provedto be optimal when tends to infinity and channel correlationlevels becomes large.

To derive the expression for , we define a new vectorthat has the same norm as the channel

vector . It is easy to see that and. As shown in [20], the expression of

can be obtained after the following steps of manipulations.First, the probability density distribution (PDF) of the channelnorm , denoted as , and the conditionalPDF for any are derived (for the de-tails please refer to [20]). Based on that, each element in canbe computed, with the non-zero diagonal elements expressed as(5), shown at the bottom of the page, 1 where the PDF isgiven by

(6)

In [19] and [21], it is proposed to use . Here, in-stead, we propose to use some extra feedback bits to obtain abetter estimate, using a channel phase coodebook. The details

1All the non-diagonal elements can be proved to be zero.

(5)

1220 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 3, MARCH 2011

of how to obtain this codebook is described in Section IV. Foreach channel realization, the user terminal selects a preferredcodeword and feeds back its index in the codebook to the basestation. If we express the codebook as ,with denoting the size of the codebook, the codeword selec-tion scheme is defined by

(7)

where denotes the Frobenius norm of a matrix. Then,we choose the preferred channel phase matrix as .Note that this can be viewed as using an adaptive codebook

instead of a fixed codebook, to quantize the channel phasematrix, thus better exploiting the information in .

Note that compared to the simple MMSE channel estimatein [19]–[21], in the proposed algorithm the limited feedbackparameter based on the phase codebook can explicitly improvethe quality of the instantaneous channel phase information inthe estimated channel representation, and thereby can reducethe risk of severe phase mismatching between the actual andestimated channel representation, especially when the channelcorrelation level is not strong enough.

Now we focus on the computational complexity of the pro-posed feedback parameter definition and channel reconstructionalgorithm. Compared to conventional methods based on channelquantization vector codebook, the additional complexity of ourscheme mainly involves the computation of both at the basestation and the user terminal. While compared to the method in[21], in our scheme each user should additionally computeand search the codebook to determine the preferred codeword.It is easy to see that the complexity of the codebook search ac-cording to the codeword selection scheme in (7) increases lin-early with and is acceptable especially for small codebooksizes such as . As for the computational complexity in-volved in , it is shown in [19] that the complexity is small ifthe following tricks are used:

1) the denominators of on the right side of (5) can beprecomputed since they are independent of ;

2) for each , the normalization factor in (5) can becomputed via the property , which yields

. Therefore, onceare obtained, can be readily obtained instead of usingits expression in (6).

C. Generalize to Multi-Antenna Receiver

It will be shown in Section VI, the channel representationreconstructed by the BS through the above method can beused for SDMA transmit optimization. However, the abovediscussion is limited to single-antenna receivers. Therefore, it isuseful to consider the possible generalization to multi-antennareceivers. Here we focus on single-stream transmission foreach active user and denote the th user’s channel matrix as

, with being the number of receive an-tennas. Note that compared to the multiple-input single-outputchannel (MISO) , it is difficult to effectively estimate themultiple-input multiple-output (MIMO) channel with the

same feedback overhead, due to the increased dimension in. Herein, we propose an alternative method to generalize the

previously proposed algorithm, using an idea similar to thosein [14] and [24].

If we predetermine a virtual receive beamforming vectorfor each user, denoted as , the BS side can see theMIMO channel of each user as a virtual MISO channel definedas . Here, this virtual beamforming vector isonly used to determine the feedback parameters and optimizethe SDMA transmission. At the receiving stage, an enhancedreceive beamforming vector can be employed to potentiallyimprove the receive performance. Based on this virtual MISOchannel, it is straightforward to generalize the proposedchannel reconstruction algorithm to multi-antenna receivers,by replacing with the virtual channel vector . Whatremains is to properly derive the virtual beamforming vectors.The important thing is that the virtual beamforming shouldbe optimized such that the correlation statistics of can bedeterministically derived both at the BS and the user terminal.

Consider the Kronecker-structured channel model given by

(8)

where the elements of are independently and identicallydistributed with and denote the transmitand receive correlation matrices, respectively, with their index

omitted for notation brevity. Typically, it can be assumedin scenarios with rich scattering at the receiver

side, this is commonly done for mathematical tractability. Sim-ilar to the method in [25], is decomposed as

(9)

where denotes a diagonal matrixwith eigenvalues arranged in a predefined order, denotesthe eigenvector with the largest eigenvalue,denotes the eigenvectors with eigenvalues comparable to thelargest one, and denotes the eigen-vectors with relatively small eigenvalues that can be neglected.Clearly, the user is most sensitive to interference in the spacespanned by . Since, in this stage, the user has no exactknowledge of its transmit beamforming and interference fromother users, it is suitable to assume that he transmits signal inthe dominant eigendirection of , and experiences interfer-ence from . Then, as one of practical solutions, the virtualreceive beamforming vector can be designed to eliminate inter-ference from and maximize its effective power, given by

(10)

Correspondingly, the correlation matrix of thevirtual channel vector is calculated as

, whichis known to both sides of the BS and the user terminal.

HUANG et al.: EXPLOITING LONG-TERM CHANNEL CORRELATION IN LIMITED FEEDBACK SDMA 1221

IV. CODEBOOK CONSTRUCTION

Obviously, the construction of channel phase codebook forchoosing a proper plays a key role in the design of limitedfeedback SDMA scheme. In this section a new method of code-book construction will be proposed. Before that, it is useful toderive the asymptotically optimal phase matrix.

A. Optimal Phase Matrix

To minimize the estimation error of channel reconstruction,the optimal phase matrix for a given is given by

(11)

where denotes the space of all the -order unitary ma-trices. Using the definition of Frobenius norm, the objective inthe above optimization problem can be simplified as

(12)

where denotes the matrix trace. In general, it is not easy togive a closed-form solution of the above optimization problem.Nevertheless, it is useful to investigate its solution in the specialcase where the channel norm is large. We start with the eigen-decomposition of , expressed as

(13)

where denotes the unitary matrix formed by the eigenvec-tors of , and is a diagonal matrix with eigenvaluesarranged in a non-increasing order, in particular here

. When the channel norm tends to infinity,it is revealed in [20] that has the asymptotic behaviors of

and for any . It follows that theobjective in (12) is asymptotically upper bounded as

(14)

The equality holds if , meaning that the optimal phasematrix is given by

(15)

B. Codebook Construction

Next, we focus on the codebook construction with finite-ratefeedback. Compared to the conventional channel quantiza-tion codebook construction, some additional issues should beaddressed in our system. First, due to the fact that the phasecodebook consists of a set of unitary matrices instead of vec-tors, the conventional method of Grassmannian subspace/linepacking [26], [27] cannot be applied here. We thus use thegeneralized Lloyd algorithm [28]–[31] to construct the phasecodebook, which is able to efficiently take the user channel’sfading correlation into account. Second, it is important tonote that, in practice, the channels of different users in a celltypically shows different correlation matrices. To fully exploit

the information in all the correlation matrices, a user-dependentphase codebook is required. However, this is not viable inpractice due to high complexity. Alternatively, we propose toconstruct a uniform codebook for all the users in a cell andadaptively update it for each specific user by . Thisway can equivalently realize a correlation-dependent codebookconstruction and has the advantage of low complexity.

To construct , we first need to define an average distortionmetric to measure the codebook performance. Since we aim toreconstruct at the base station, it is reasonable to define thefollowing distortion metric:

(16)

In order to iteratively improve the codebook over the th can-didate , the Lloyd algorithm generatesa large set of channel realizations referred as trainingsequence (TS), where denotes the size of TS. The corre-sponding fading correlation matrices are denoted as .Note that in theory should span all the possible fadingcorrelation realizations for a particular BS configuration. How-ever, in practice, a finite set of representative matrices areemployed for simplicity. The updated codebook can then be de-rived to minimize the approximate distortion metric, expressedas

(17)

where is the partition region of TS associated to codeword, and

(18)

with being the eigenvectors of , and being a functionof and .

The main difficulty of the above Lloyd algorithm lies infinding a solution of (17), which can be simplified to a ele-ment-wise optimization, given by

(19)

A quasi-optimal solution (in-dices are omitted for notational simplicity) for the aboveoptimization problem is obtained as

(20)

where , with denotingthe th diagonal element of is the eigenvectors of

1222 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 3, MARCH 2011

TABLE ICODEBOOK SAMPLES WITH� � �

with the zero-valued eigenvalues, i.e.,is a set of orthogonal vectors spanning the null space of

, and denotes the eigenvector of the matrix withthe largest eigenvalue. For a proof, please refer to Appendix A.

Note that the above generalized Lloyd algorithm can not guar-antee a convergence to the global optimality. But it provides apractical way of codebook design. Because the codebook can beconstructed offline, the complexity of the above algorithm willnot become an issue. In order to increase the probability of ap-proaching the global optimality, the above generalized Lloyd al-gorithm can be attempted many times with different initial code-books, using the generated codebook with the smallest averagedistortion as the final one.

Table I shows some codebook samples generated using theproposed method, where the channel correlation of the givenmultiuser network is calculated as a function of the distancebetween the neighboring antennas and the angle spread. Forthe detailed configuration, please refer to the simulation setupin Section VII. It is shown that the average distortion decreaseswith increasing codebook size, and also decreases with de-creasing angle spread, i.e., with increasing correlation level.

V. PERFORMANCE ANALYSIS

In this section, we will first present the asymptotic perfor-mance of the proposed channel reconstruction method, and theninvestigate the effect of the phase codebook on the estimationperformance.

A. Asymptotic Analysis

Here we aim to investigate the asymptotic performance of theproposed channel reconstruction method with becoming large.Since we consider a multiuser scenario with correlated channelfading, the first eigenvalue of is in general much biggerthan the others, resulting in that the termstend to zero when is large enough. Based on these observa-tions, as shown in [20], the asymptotic behavior of can beexpressed as

(21)

The asymptotic distortion between the actual and its re-construction expression can then be calculated as

(22)

where the second equality follows from (14) and (15). We cansee that the distortion of the proposed channel reconstructionmethod asymptotically approaches a constant, and it is easilyproved that this constant decreases with the correlation levelincreasing. Especially, it tends to zero when the correlation levelis quite high.

B. Performance Loss From Codebook Quantization

Note that we use the asymptotically optimal phase matrix inthe above analysis. However, in practical use, this phase matrixhas to be quantized via the codebook introduced in Section IV,and results in some performance loss. Therefore, it is usefulto analyze the performance loss resulting from the codebookquantization with a given size . At first we need to define aperformance loss measure. As shown in Section IV-A, the per-formance of channel reconstruction is directly reflected by thevalue of that is asymptotically upper boundedby , it is therefore reasonable to define a measure as fol-lows:

(23)

which is always less than unit and indicates the ratio of theloss to the optimal performance. For a designed codebook anda given channel norm level , it is important to evaluate the av-erage loss measure over random channel realizations. For con-venience, we first define a metric of minimum distance for thedesigned codebook , given by

(24)

where denotes the first column of the codeword . Notethat this minimum distance can also be regarded as the minimumdistance on a Grassmannian line packing [26], with

being seen as a set of lines in . It is worthy tomention that this metric can not fully evaluate the phase code-book as it does not take the effect of other columns (except thefirst column) of the unitary codeword into account. Nonetheless,this metric is also effective as is typically dominated by thefirst diagonal element, and we find this metric is convenient tobe used to analyze the average performance loss. Specifically,the average loss can be upper bounded as

(25)

HUANG et al.: EXPLOITING LONG-TERM CHANNEL CORRELATION IN LIMITED FEEDBACK SDMA 1223

For a proof, please refer to Appendix B. Since the minimumdistance for the line packing in the case of isupper bounded as [26], [32]

(26)

it is easy to see that when , the average performanceloss from codebook quantization decreases with the minimumdistance for fixed .

VI. USER SCHEDULING AND MULTIUSER PRECODING

To complete the SDMA transmit scheme, in this section weshow that the reconstructed channel knowledge can beused at the BS to schedule users and design the multiuser pre-coder.

A. Greedy User Scheduling

We first focus on the user scheduling algorithm. In general,the optimal user scheduling algorithm requires an exhaustivesearch over all the possible user sets, which usually is compu-tationally prohibitive especially when the user number is large.In contrast, some suboptimal user scheduling algorithms [7], [8]have been proposed to reduce the computational complexity, in-cluding the greedy user selection (GUS) and the semi-orthog-onal user selection (SUS). Both of these two algorithms can begeneralized to our case based on the estimated channel knowl-edge . Considering the easier user in practice, we focuson GUS in the following analysis.

To optimize the rate performance, the user scheduling algo-rithm is designed to select a set (with its cardinality denotedas ) of users based on the estimated weighted sum rate givenby

(27)

where denotes the estimated signal tointerference plus noise ratio (SINR) of user , expressed as2

(28)

and denotes the weighting factor for user . Note that inorder to guarantee fairness among users especially when users

2Note that in order to reduce the risk of overestimating the SINR, an adaptiveback off margin can be added to the SINR estimate. This can be realized by com-puting the mean-square error (MSE) of the signal/interference power estimate[20], [21], denoted as ����� � � � ��. Through that, the SINR estimatecan be updated as

�� �� ��� �� � � � ��� �� � � � ��

� � � �� � � � ��� �� � � � ��

where � � � is a tuning parameter that is chosen to achieve a target error rate.However, it involves a much higher computational complexity. For this reasonwe only consider the simple SINR estimate in (28).

have different average SNRs, maximization of the weighted sumrate is a suitable criterion to use in the scheduling algorithm.The basic idea of GUS is to determine the user set using thefollowing iterative method: At iteration , a new user is addedonly if the estimated weighted sum rate with the new user set isincreased over the result at iteration .

B. Multiuser Precoding

Clearly, the multiuser precoder also plays a key role inthe above user scheduling algorithm. Fortunately, based on thechannel knowledge , some low-complexity multiuseruser precoding scheme [2]–[6] can be directly applied orgeneralized to our system. Here, we focus on the ZF multiuserprecoding scheme that has been widely used in the literatureand therefore is suitable for performance comparison.

Note that due to limited feedback constraint, both the userselection and multiuser precoding are based on the estimatedchannel knowledge . As shown in Section III-B, is notstrictly restricted to be rank-one in order to provide robustnessagainst extreme channel realizations. Therefore, if following theconventional ZF criterion, the estimated interuser interference

should be forced to be zero, which is toostrict and may destroy the effectiveness of GUS. For this reason,we would like to generalize the ZF scheme in a way similar to[21].

It is shown that, in general, is of full rank but dominatedby a few number of eigenmodes. So we can define its domi-nating subspace which is sensitive to the multiuser interferenceand use it in the precoding design. If we rewrite in (2) as

(29)

where is defined as the dominating eigenvectors with theircorresponding eigenvalues satisfying(in which is a threshold that is always less than one). Thespace spanned by the dominating eigenvectors is then definedas the dominating subspace in which the user is sensitive to in-terference. Base on that, the generalized ZF scheme is designedto restrict the user signal in the null space of other active users’dominating subspaces, mathematically given by

(30)

To solve the optimization of , we first define a projectionmatrix as

(31)

where denotes a column orthonormal matrix, which is ob-tained by applying the Gram-Schmidt orthonormalization to thecolumns in . The beamforming can then becalculated as

(32)

where denotes the first column of , and is a scalingused to fulfill the transmit power constraint.

1224 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 3, MARCH 2011

VII. SIMULATION RESULTS

In this section, computer simulations will be performed toevaluate the proposed SDMA scheme under different levels ofcorrelated channel environments. We consider a single-cell mul-tiuser network, where the base station is equipped with a uni-form linear array (ULA) of transmit antennas and allthe users are randomly distributed in a circle around the base sta-tion. The channels are assumed to be block flat fading, that is, thechannels keep static during one burst of user scheduling and datatransmission but vary independently between different bursts.For channel realization, the channel vectors are generated using

, where is a vector whose entries are indepen-dently and identically distributed with , and denotesthe spatial correlation matrix, which is computed by assuminga Gaussian distributed scattering with the AoD as the mean andwith the angle spread as the standard deviation [33]. In par-ticular, AoDs of different users are randomly distributed be-tween 60 and 60 degrees, meaning that the generation ofis user dependent. The distance between two neighboring an-tennas at the antenna array is set to be four carrier wavelengths.Throughout the simulations, we choose for all . Theaverage SNR in the simulations is defined as ,and the calculation of the sum rate assumes a perfect rate adap-tion. The channel phase codebook is constructed by attempting20 times with different initial values and usingthe one with the smallest average distortion as the final one.

For performance comparison, we also simulate some state ofthe art limited feedback SDMA schemes that can also exploit thelong-term channel correlation information, listed as follows.

• The scheme based on generalized ZF precoding (GZF) andMMSE estimate of the channel representation, in whichthe feedback scalar only includes a single channel norm[19]–[21].

• The scheme based on ZF precoding and channel vectorquantization [14], in which a channel quantization code-book is designed for arbitrary spatial correlation statistics,also using the Lloyd algorithm.

• The scheme based on the estimate of user-experienced in-terference [18], in which the exhaustive user selection isused, and the feedback scalars include the channel normand the alignment parameter (AP) between the normalizedchannel vector and the preset beamforming vector.

• The scheme based on a precoding codebook [11] proposedby Samsung corporation, in which each user chooses a pre-ferred beamforming vector in the codebook and feeds backits index as well as the corresponding SINR.

Note that if not stated explicitly, the above schemes employthe same user selection algorithm as the proposed one.

Figs. 1 and 2 illustrate the cumulative distribution func-tion (CDF) of the rate achieved by the limited feedback SDMAschemes. Results show that in the configuration and

dB, the proposed scheme outperformsothers in both correlated channels with and .If we define the outage rate as ,where denotes the actual achievable rate, and denotes theoutage probability. For a typical outage probability ,it is seen from both figures that with the same feedback amount,

Fig. 1. The CDF of the sum rate achieved by the SDMA schemes,� � ��

� � �� � � ��� ��� � 15 dB.

Fig. 2. The CDF of the sum rate achieved by the SDMA schemes, � � ��

� � ��� � � ��� ��� � 15 dB.

i.e., a CQI (channel norm or SINR) plus a 3-bit preferred code-word indicator (PCI), the outage rate of the proposed schemehas a gain of around 1.5 b/s/Hz over the Samsung scheme.Compared to other existing schemes, the proposed schemeexhibits a gain of more than 2 b/s/Hz. This result confirms theeffectiveness of the proposed channel reconstruction method.

The average rate of the proposed scheme as well as thereference schemes are evaluated in Figs. 3–6. It is seen that,compared to the GZF scheme based on MMSE estimate, theproposed scheme shows a performance advantage of around1 b/s/Hz at a wide SNR and user-number region. The gain isslightly enhanced with the correlation level decreasing from

to . This is because in the proposed schemean additional PCI is used to improve the channel quality at theBS. Compared to the schemes based on channel vector quan-tization codebook and interference estimate, results show thatwith the same feedback overhead, the proposed scheme shows

HUANG et al.: EXPLOITING LONG-TERM CHANNEL CORRELATION IN LIMITED FEEDBACK SDMA 1225

Fig. 3. The average rate of the SDMA schemes varying with SNR,� � ��

� � �� � � ��.

Fig. 4. The average rate of the SDMA schemes varying with SNR, � � ��

� � ��� � � ��.

considerable gain especially at medium–high SNR region andsmall–medium number of users. In contrast to the Samsungscheme, the proposed scheme only shows advantage at highSNR region and small number of users. Note that the Samsungscheme is designed to directly quantize the multiuser precodingmatrix via codebook construction. This strategy usually has agood estimate of the sum rate for each candidate user set butmay experience precoding mismatch at small number of users.Alternatively, the proposed scheme and other state of the artschemes listed above aim to estimate the sum rate based on thereal-time derived multiuser precoding matrix, which reducesthe risk of precoding mismatch but may experience a poorestimate of the sum rate at the BS side. This strategy differencecan explain the above comparing result between the proposedscheme and the Samsung scheme.

In order to evaluate the impact of the channel correlationlevel on the performance of the proposed SDMA scheme, Fig. 7

Fig. 5. The average rate of the SDMA schemes varying with the number ofusers, � � �� � � �� ��� � 15 dB.

Fig. 6. The average rate of the SDMA schemes varying with the number ofusers, � � �� � � ��� ��� � 15 dB.

shows the average rate of the SDMA schemes varying with theangle spread. It is known that the channel correlation level de-creases with the angle spread increasing. Results show that therate performance of all the schemes decrease with the correla-tion level, that is because they are all able to exploit the channelcorrelation statistics and thereby desire for a higher correlationlevel. To compare, it is seen in the configuration

and 15 dB, the proposed scheme shows a gainof at least 0.5 b/s/Hz over other schemes at , and hasmore gains at a larger .

VIII. CONCLUSION

This paper has proposed an SDMA scheme that combineslimited feedback with long-term channel statistics. In order toacquire a good quality of CSI at the BS, this proposed scheme isdesigned to reconstruct the outer product of each channel vectorby itself, instead of aiming to directly quantize the channel

1226 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 3, MARCH 2011

Fig. 7. The average rate of the SDMA schemes varying with the angle spread,� � �� � � ��� ��� � 15 dB.

vectors. This is realized by a newly developed channel phasecodebook and the eigendecomposition-structured channel re-construction method. Using this form of channel knowledge,the BS can complete the user scheduling and transmit beam-forming optimization. Computer simulation results have shownthat, under the correlated channel environment, the proposedscheme mostly outperforms state of the art schemes in varioussystem configurations.

APPENDIX APROOF OF (20)

If we expand the expression of and asand re-

spectively, the objective function in (19) can be rewritten as

(33)

Since it is difficult to analytically solve this problem, we wouldlike to optimize column by column, using a greedy algorithm.

To be more specific, we first optimize such that the first termin the summation of the objective function is maxi-

mized, which is given by

(34)

After that, we then optimize and use the second term of thesummation in (33) as the objective function. In the meanwhile,we should make sure that is orthogonal to the already opti-mized . Following this way, all the columns in can be op-timized step by step, and the optimization of for anycan be expressed as

(35)

It is clear that the ordering of the column optimizations inthe above greedy algorithm has an impact on the final perfor-mance. The reason we choose to optimize the columns offrom the left to the right is from the observation

. The optimization problem in (34) is an eigen-value problem and can be solved using a standard approach,and the problem in (35) can also be transformed into a stan-dard eigenvalue problem. In particular, if we define a new ma-trix , the second constraint in (35) can berewritten as , which means that belongs tothe null space of , i.e., the space spanned by theeigenvectors (denoted as henceforth) of correspondingto the zero-valued eigenvalues. Thus, the can be rewritten as

where is an arbitrary vector with unit norm. Now the opti-mization with respect to can be transformed to a new opti-mization problem with respect to , given by

(36)

which is obviously a standard eigenvalue problem and the op-timized can be expressed as . This concludesthe proof.

APPENDIX BPROOF OF (25)

We collect the first columns of all the codewords in-cluded in into a set and denote it as a new codebook

. Obviously, its minimum distance

HUANG et al.: EXPLOITING LONG-TERM CHANNEL CORRELATION IN LIMITED FEEDBACK SDMA 1227

is equal to defined in (24). Following (23), the perfor-mance loss can be further expressed as

(37)

where and is the normalization of . Usingthe idea behind the beamforming codebook design in [23], [26],

can be viewed as a set of lines in the so-called Grassmannmanifold, rather than a set of vectors, and is viewed as arandom point in the same Grassmann manifold. Then, based onthe theory about the density of a line packing [26], we have

(38)

where denotes the density of the line packing, given by

(39)

Thus, the upper bound of the average performance loss can bewritten as

(40)

The inequality in the second line is obtained by observing thatthere are two cases of the channel realization correspondingto if the condition issatisfied. When the condition is not fulfilled (with probability

as shown in (38)), we use the trivial bound. This concludes the proof.

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Yongming Huang (M’10) received the B.S. andM.S. degrees from Nanjing University, China, in2000 and 2003, respectively, and the Ph.D. degreein electrical engineering from Southeast University,China, in 2007.

After graduation, he held teaching/research po-sitions at the School of Information Science andEngineering, Southeast University, China, wherehe is currently an Associate Professor. During2008–2009, he was visiting the Signal ProcessingLaboratory, Electrical Engineering, Royal Institute

of Technology (KTH), Stockholm, Sweden. His current research interestsinclude MIMO communication systems, multiuser MIMO communications,cooperative communications, and satellite mobile communications.

Luxi Yang (M’96) received the M.S. and Ph.D. de-gree in electrical engineering from the Southeast Uni-versity, Nanjing, China, in 1990 and 1993, respec-tively.

Since 1993, he has been with the Department ofRadio Engineering, Southeast University, where heis currently a Professor of information systems andcommunications, and the Director of the DigitalSignal Processing Division. His current researchinterests include signal processing for wireless com-munications, MIMO communications, cooperative

relaying systems, and statistical signal processing. He is the author or coauthorof two published books and more than 100 journal papers, and holds ten patents.

Prof. Yang received the first- and second-class prizes of the Science and Tech-nology Progress Awards of the State Education Ministry of China in 1998 and2002. He is currently a member of the Signal Processing Committee of the Chi-nese Institute of Electronics.

Mats Bengtsson (M’00–SM’06) received the M.S.degree in computer science from Linköping Univer-sity, Linköping, Sweden, in 1991 and the Tech.Lic.and Ph.D. degrees in electrical engineering from theRoyal Institute of Technology (KTH), Stockholm,Sweden, in 1997 and 2000, respectively.

From 1991 to 1995, he was with Ericsson TelecomAB Karlstad. He currently holds a position as Asso-ciate Professor in the Signal Processing Laboratoryof the School of Electrical Engineering at KTH. Hisresearch interests include statistical signal processing

and its applications to antenna-array processing and communications, radio re-source management, and propagation channel modeling.

Dr. Bengtsson served as Associate Editor for the IEEE TRANSACTIONS ON

SIGNAL PROCESSING from 2007 to 2009 and is a member of the IEEE SPCOMTechnical Committee.

Björn Ottersten (S’87–M’89–SM’99–F’04) wasborn in Stockholm, Sweden, in 1961. He receivedthe M.S. degree in electrical engineering and appliedphysics from Linköping University, Linköping,Sweden, in 1986 and the Ph.D. degree in electricalengineering from Stanford University, Stanford, CA,in 1989.

He has held research positions at the Department ofElectrical Engineering, Linköping University; the In-formation Systems Laboratory, Stanford University;and the Katholieke Universiteit Leuven, Leuven, Bel-

gium. During 1996 to 1997, he was Director of Research at ArrayComm, Inc.,San Jose, CA, a start-up company based on his patented technology. In 1991,he was appointed Professor of Signal Processing at the Royal Institute of Tech-nology (KTH), Stockholm. From 2004 to 2008, he was Dean of the School ofElectrical Engineering at KTH, and from 1992 to 2004 he was head of the De-partment for Signals, Sensors, and Systems at KTH. Since 2009, he has beenDirector of securityandtrust.lu at the University of Luxembourg. His researchinterests include wireless communications, stochastic signal processing, sensorarray processing, and time series analysis.

Dr. Ottersten is a first recipient of the European Research Council advancedresearch grant. He has coauthored papers that received an IEEE Signal Pro-cessing Society Best Paper Award in 1993, 2001, and 2006. He has served asAssociate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING and onthe Editorial Board of the IEEE SIGNAL PROCESSING MAGAZINE. He is currentlyEditor-in-Chief of the EURASIP Signal Processing Journal and a member of theEditorial Board of the EURASIP Journal of Advances Signal Processing. He isa Fellow of EURASIP.